The city of Warsaw, never known as cycling paradise, has taken a step that Budapest's authorities have stubbornly resisted: allowing bicyclists to use bus lanes.
In Budapest (and throughout Hungary) the default rule is that bicycles cannot ride in bus lanes on the grounds that the lanes are too narrow to safely accommodate both modes of transport (nevermind that taxis have carte blanche here).
On rare streets where there is an abundance of width, authorities, if they choose, can post bike signs and/or mark bike lanes that allow for an exception.
However, on most bus priority streets downtown (e.g. on Árpád fejdelem and an Erzsébet híd-Rákóczi út), buses hold sway in the curb lane and cyclists are required to ride in the second lane over, where they're being passed by buses on their right and other traffic on the left. Here's an illustrative video.
In Warsaw, authorities have recognised the absurdity of this situation. Whatever challenges might be posed by opening bus lanes to cyclists, it's more dangerous to force them between lanes of faster-moving traffic. In addition, the Warsaw authorities reasoned that because cyclists make up a relatively small share of traffic (as in Budapest, like it or not), the disruption to bus traffic will probably be tolerable. Warsaw will therefore open one bus lane to cyclists and other single-track vehicles on the trial basis, and then expand the idea depending on results.
In Budapest, so far, they've been very dogmatic about lane-width requirements -- even to the point of creating a dangerous situation for cyclists. The city should follow Warsaw's common-sense approach, and see if a more permissive policy works on an experimental basis.